On The Computational Cost of FFT-Based Linear Convolutions

The “linear convolution” of two n-long sequences x and y is commonly performed by extending the input sequences with zeroes to length 2p, where p is the smallest power of two greater than or equal to n, and then evaluating the circular convolution of these sequences using power-of-two FFTs. This approach clearly favors power-of-two input sequences sizes, with abrupt increases in computational cost when the size exceeds a power of two. In this article, a recursive technique is presented for efficiently computing linear convolutions for any size n, even if one uses power-of-two FFTs as the underlying computational engine. The computational cost for this technique is never greater than the conventional approach and usually significantly less. Further, the computational cost as a function of n is highly continuous, so that linear convolutions of sizes somewhat larger than a power of two, for example, are only slightly more expensive than linear convolutions of power-of-two length data. D. Bailey: NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, CA 94035-1000. E-mail: [email protected].